Page 29 - Textos de Matemática Vol. 45

2.3. First properties of Schreier and homogeneous split epimorphisms 21 (b) the commutative square f′u = vf is a pullback square;
(c) the upper row, with the map q = K(u)−1q′u, is a Schreier split epimor- phism.
Proof. (a)⇒(b) Given l: Z →A′ and g: Z →B, such that f′l = vg, we can define the map h: Z → A by
h(z) = K(u)−1q′l(z) · sg(z), z ∈ Z. Then uh = l and fh = g, indeed
u(K(u)−1q′l(z) · sg(z)) = q′l(z) · s′vg(z) = q′l(z) · s′f′l(z) = l(z), and
f(K(u)−1q′l(z) · sg(z)) = g(z).
Moreover, h is a morphism: in order to show this fact, since u and f are jointly monomorphic, it suffices to show that uh(z·z′) = uh(z)·uh(z′) and fh(z·z′) = fh(z) · fh(z′). We have that
u(K(u)−1q′l(z · z′) · sg(z · z′)) = q′(l(z) · l(z′)) · s′vg(z) · usg(z′)
= q′l(z) · q′(s′f′l(z) · q′l(z′)) · s′vg(z) · usg(z′)
= q′l(z) · q′(s′f′l(z) · q′l(z′)) · s′f′l(z) · usg(z′) = q′l(z) · s′f′l(z) · q′l(z′) · usg(z′)
= u(K(u)−1q′l(z) · sg(z) · K(u)−1q′l(z′) · sg(z′)) = uh(z) · uh(z′)
fh(z · z′) = f(K(u)−1q′l(z · z′) · sg(z · z′)) = g(z) · g(z′) =
and
The uniqueness required for the universal property of the pullback follows from the fact that the pair (u,f) is jointly monomorphic.
(b)⇒(c) follows from Proposition 2.3.4.
(c)⇒(a) Consider l,g: Z →A with fl = fg and ul = ug. If (f,s) is a
Schreier split epimorphism with the map q = K(u)−1q′u then we have l(z) = ql(z) · sfl(z)
and
g(z) = qg(z) · sfg(z).
Since fl = fg, in order to prove that l = g it suffices to show that ql(z) = qg(z), but this follows directly from the fact that q = K(u)−1q′u and ul = ug.
= f(K(u)−1q′l(z) · sg(z) · K(u)−1q′l(z′) · sg(z′)) = fh(z) · fh(z′).